Margulis Lemma, entropy and free products

We prove a Margulis' Lemma à la Besson-Courtois-Gallot, for manifolds whose fundamental group is a nontrivial free product A ∗ B, without 2-torsion. Moreover, if A ∗ B is torsion-free we give a lower bound for the homotopy systole in terms of upper bounds on the diameter and the volume-entropy. We also provide examples and counterexamples showing the optimality of our assumption. Finally we give two applications of this result: a finiteness theorem and a volume estimate for reducible manifolds.